Mean Field Interactions in Heterogeneous Networks

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Abstract

In the context of complex networks, we often encounter systems
in which the constituent entities randomly interact with each other as they
evolve with time. Such random interactions can be described by Markov processes,
constructed on suitable state spaces. For many practical systems (e.g. server farms,
cloud data centers, social networks), the
Markov processes, describing the time-evolution of their constituent entities,
become analytically intractable
as a result of the complex interdependence
among the interacting entities.
However, if the `strength' of
these interactions converges to a constant as the size of the system
is increased, then
in the large system limit the underlying Markov process converges to a deterministic process, known
as the mean field limit of the corresponding
system. Thus, the mean field limit
provides a deterministic approximation of the randomly evolving system.
Such approximations are accurate for large system sizes.
Most prior works on mean field
techniques have analyzed systems in which the constituent entities
are identical or homogeneous.
In this dissertation, we use mean field techniques to analyze large complex systems
composed of heterogeneous entities.
First, we consider a class of large multi-server systems, that
arise in the context of web-server farms and cloud data centers. In such systems,
servers with heterogeneous capacities work in parallel to process
incoming jobs or requests.
We study schemes to assign the incoming jobs to the servers
with the goal of achieving optimal performance in terms of
certain metrics of interest while requiring the state information
of only a small number of servers in the system.
To this end, we consider randomized dynamic job assignment schemes
which sample a small random subset of servers at every job arrival instant
and assign the incoming job to one of the sampled servers based
on their instantaneous states.
We show that for heterogeneous systems,
naive sampling of the servers may result in an `unstable' system.
We propose schemes which maintain stability
by suitably sampling the servers.
The performances of these schemes are studied via the corresponding mean field limits,
that are shown to exist.
The existence and uniqueness of an asymptotically stable
equilibrium point of the mean field is established in every case.
Furthermore, it is shown that, in the large system limit,
the servers become independent of each other and the stationary
distribution of occupancy of each server can be obtained from the unique
equilibrium point of the mean field. The stationary tail distribution
of server occupancies is shown to have a fast decay rate which suggests significantly
improved performance for the appropriate metrics relevant to the application. Numerical studies
are presented which show that the proposed randomized dynamic schemes significantly outperform
randomized static schemes where job assignments are made independently
of the server states. In certain scenarios, the randomized dynamic schemes are observed
to be nearly optimal in terms of their performances.
Next, using mean field techniques, we study
a different class of models
that arise in the context of social networks.
More specifically, we study the impact of social interactions
on the dynamics of opinion formation in a social network
consisting of a large number of interacting social agents.
The agents are assumed to be mobile and hence do not have any fixed set of
neighbors.
Opinion of each agent is treated as a binary random variable,
taking values in the set {0,1}. This represents scenarios,
where the agents have to choose from two available options.
The interactions between the agents are modeled using
1) the `voter' rule and 2) the `majority' based rule.
Under each rule, we consider two scenarios, (1) where the agents
are biased towards a specific opinion and
(2) where the agents have different propensities to change
their past opinions.
For each of these scenarios, we characterize the
equilibrium distribution of opinions in the network
and the convergence rate to the equilibrium
by analyzing the corresponding mean field limit.
Our results show that the presence of biased agents can significantly
reduce the rate of convergence to the equilibrium. It is also observed that,
under the dynamics of the majority rule, the presence of `stubborn' agents
(those who do not update their opinions)
may result in a metastable network,
where the opinion distribution of the non-stubborn agents
fluctuates among multiple stable configurations.